I would trust the liquid test result over the strip test result.
Unless its coming from your source water, high nitrate is the result of high amounts of ammonia. Ammonia is being converted into nitrate.
Typically, a normally stocked tank will produce around 1ppm of ammonia per day. The nitrogen cycle will convert this into nitrate and you would see nitrate rise by about 4ppm per day. A 30 to 50% weekly water change with clean (0 nitrate) would be sufficient to keep nitrate at low levels in these circumstances.
If your tank is producing higher levels of ammonia than this you will see higher levels of nitrates with the same water change routine. Similarly, if your source water has high levels of nitrate in it, the water change wont bring the nitrate down by the same amount and you will see higher levels of nitrate.
Have you tested your tap water as suggested?
Have these high levels of nitrate suddenly appeared?
Your fish would be producing low levels of ammonia in say a 50g tank, but those same fish in 10g would have 5x higher ammonia levels because its being diluted 5x less. What size tank do you have?